Solutions to two problems in component theory of complete lattices
Pei Dao · Journal of Shaanxi Normal University · 2000
The following two open problems in the component theory of complete lattices are studied: (1) In a non distributive complete lattice, is it true that any component of an element with finite width is also finite wide? (2) What functions between complete lattices can preserve the granule representable property or granule property? Through constructing a counterexample it is showed that there is a non distributive complete lattice in which a component of some element with finite width is not finite wide. Furthermore, it is showed that an isomorphism between complete lattices can preserve the granule representable property and granule property, and counterexamples are given to show that the condition of isomorphism can not be weakened. So the two open problems list above are completely solved.